Contents
Does the Fourier transform always exist?
There is never a question of existence, of course, for Fourier transforms of real-world signals encountered in practice. However, idealized signals, such as sinusoids that go on forever in time, do pose normalization difficulties.
What are the conditions for existence of Fourier transform?
The conditions are: f must be absolutely integrable over a period. f must be of bounded variation in any given bounded interval. f must have a finite number of discontinuities in any given bounded interval, and the discontinuities cannot be infinite.
What are the limitations of Fourier transform?
The major disadvantage of the Fourier transformation is the inherent compromise that exists between frequency and time resolution. The length of Fourier transformation used can be critical in ensuring that subtle changes in frequency over time, which are very important in bat echolocation calls, are seen.
What is the value of J in Fourier transform?
The letter j here is the imaginary number, which is equal to the square root of -1. I will use j as the imaginary number, as is more common in engineering, instead of the letter i, which is used in math and physics. This exponential representation is very common with the Fourier transform.
What is the first Dirichlet condition?
Explanation: In the case of Dirichlet’s conditions, the first property leads to the integration of signal. It states that over any period, signal x(t) must be integrable.
What is J in Fourier?
But can you please me what the >term ‘j’ stands for in the Fourier transform when we multiply our signal >(be it in time or frequency domain) by an imaginary/complex exponential >function. j*j = -1 or j is the complex number with unit magnitude and real part equal to zero.
Can you get the Fourier transform by replacing s with J?
In general you can’t get the Fourier transform by replacing s with j ω and vice versa. Two conditions must be satisfied in order for this to lead to a correct result: Both transforms must exist (in the sense that the corresponding signal x ( t) has a Laplace transform and a Fourier transform).
Do you have to have both Laplace transform and Fourier transform?
Both transforms must exist (in the sense that the corresponding signal x ( t) has a Laplace transform and a Fourier transform). The imaginary axis s = j ω must be inside the region of convergence of the Laplace transform.
How to convert Fourier transform to Omega form?
You get the two following basic Fourier transform correspondences, depending on whether you use f or ω = 2 π f as the frequency domain variable: So your expression for X ( ω) misses a factor of π, and the sums and differences of the two frequencies in the arguments of the Dirac deltas need to be multiplied by 2 π:
What’s the difference between X ( J Omega ) and X ( s )?
$X(j omega)$ (frequency response) is a Fourier transform of system’s impulse response. It’s actually a function of frequency ($omega$) but usually is written as $X(j omega)$ because replacing $j omega$ in the formula with $s$ will give you system’s Laplace transform $X(s)$ without any additional conversions.